The K\"ahler-Ricci flow and the ∂ operator on vector fields
Abstract
The limiting behavior of the normalized K\"ahler-Ricci flow for manifolds with positive first Chern class is examined under certain stability conditions. First, it is shown that if the Mabuchi K-energy is bounded from below, then the scalar curvature converges uniformly to a constant. Second, it is shown that if the Mabuchi K-energy is bounded from below and if the lowest positive eigenvalue of the ∂ ∂ operator on smooth vector fields is bounded away from 0 along the flow, then the metrics converge exponentially fast in C∞ to a K\"ahler-Einstein metric.
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